' """v‘o* APPLiCATION as THE 1115an OF mamas TO THE svaEszs OF MECHANISMS Thesis for the Degree of Ph. D. MECHIGAN STATE UMVERSITY‘ WilEiam H. Busseil i964 TH ESl3 .0 . ’2 if] \. ABSTRACT APPLICATION OF THE THEORY OF ROULETTES TO THE SYNTHESIS OF MECHANISMS by William H. Bussell Four bar mechanisms are desirable machine elements because of their simplicity. However, because the proper- ties of such a mechanism are changed when the relative lengths of its links are changed, out and try methods of designing them are time consuming. Graphical methods are useful and help to give a feeling for the mechanism. Ana- lytical methods, however, provide means of programming a digital computer for a numerical solution. The approach used here in devising an analytical solution, is that of instantaneous motion of the coupler bar plane and as such belongs to the class of analyses based on infinitesimal displacements. The theory of rou- lettes, which treats of the paths of points in the plane of a curve rolling without slipping on another, is used. This supplies a means of applying the concept of station- ary curvature of point paths in obtaining a numerical solution to a mechanism synthesis problem. Since any plane motion can be reduced to the motion of a curve rolling on another Curve, part of the problem is one of determining a suitable rolling curve pair. ‘William H. Bussell The instant center concept is used to obtain the rolling curve pair for a given path or function generation problem. The equations of the rolling curve pair are then used with principles from the Calculus to determine the location of all points in the moving plane which, during the instantaneous motion chosen, move in paths of station- ary curvature. Any two of these points are used as hinge joints at one end of a pair of links joining the moving plane to the fixed plane. The links are joined to the fixed plane at the centers of curvature of the pair of points chosen. This forms a four bar mechanism. The plane motion of the coupler bar of this mechanism will closely approx imate the motion of the moving curve plane over a small range of displacement. This method is useful in devising mechanisms in which a point on the coupler bar traces a portion of some required continuous curve. It is also useful, by means of mechanism inversion, for devising function generator mech- anisms. If the function generator can be made with a pair of rolling curves, a portion of the motion of the rolling curve can be generated with a four bar mechanism. An analytical method of determining the output angle of the function generator of this mechanism is devised so that a computer can be programmed to test possible solu- tions of a given problem. The method does not supply the dimensions of the best linkage arrangement, so there William H. Bussell remains the problem of testing a finite number of possible mechanisms in order to obtain one which will satisfy the problem. APPLICATION OF THE THEORY OF ROULETTES TO THE SYNTHESIS OF MECHANISMS By William H. Bussell A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1964 PREFACE The synthesis of mechanisms has received much at- tention for many years. Before Werld war II almost all of the methods used were graphical. During the war years the increased use of mechanical analog computers and function generators focused more attention to the need for more ac- curate methods of synthesis. Since then, and particularly during the last ten years, various analytical methods have been developed for the synthesis of the basic four bar linkage. While the four link mechanism is simple in appear- ance in that there are only three moving links, the analy- sis of the motion of the linkage is not simple. There are many theories and techniques in use. One of these, refer- red to later as the inflection circle concept, has been in use for many years and a special terminology has been built up around it. However, there seems to be no strict- ly analytical method of synthesis based on the theory underlying this method. The object of this investigation was to develOp a procedure for applying the theory of roulettes to two kinds of synthesis problems: mechanisms for tracing curves and mechanisms to generate functions. The inflec- ii th1circle concept originates in the theory of roulettes. The writer wishes to express his thanks to Dr. G. H. Martin of the Department of Mechanical Engineering for encouragement and suggestions while this work was in prep- aration. iii TABLE OF CONTENTS PREFACE . . . . . . . . . . . . . . . . LIST OF TABLES . . . . . . . . . . . . LIST OF ILLUSTRATIONS . . . . . . . . . INTRODUCTION . . . . . . . . . . . . . Chapter I. THEORETICAL DEVELOPMENT . . . . Roulettes Plane Motion and the Rolling The Point Path Traced in the II. PATH GENERATION . . . . . . . . Open Curves Closed Curves III. FUNCTION GENERATION . . . . . . Function Generators Synthesis of the Four Link Function Generator The Analytical Determination of the Output Pair Plane Crank Angle for the Four Bar Function Generator CLOSURE . . . . . . . . . . . . . . . . APPENDIX . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . iv Page ii vi 31 #8 76 80 85 LIST OF TABLES Table Page 1. Tabulated Values for the X and Y Components of Points of Stationary Curvature and Their Centers for the Straight Line Mechanism . . . . . . . . . . . . . . . . . 27 2. Results for the Parabolic Path Program . . . . 36 3. Results for the Function Generator Program . . 63 A. Computed Output for the Function Generator OfFigure 2L... 0 00.0 0-0.». 0‘. 000 O 73 Figure l. 2. 3. 10. ll. 12. 13. lb. 15. 16. 17. LIST OF ILLUSTRATIONS The Four Link Mechanism . . . . . . . . . . The Crossed Four Link Mechanism . . . . . . Three Roulettes: The Cycloid 02 and the TrOChOidS Cl and 03 o o o o o o o o o o o A Four Link Mechanism to Replace Rolling MOtoion O O O O O O O O O O I O O O O O O The Fixed and Moving Coordinate Systems . . The Location of the Fixed Coordinate System Relative to the MOving System . . . . . . The Disc and Straight Line Rolling Pair . . The General Tracing Point and the Rolling Curve Pair 0 O 0 O O O O O O O O O O O O O O O The Rolling Curve Pair and the Derived Approx- imating Four Link Mechanism . . . . . . . The Locus Curves for the Straight Line Mech— anism.................. The Straight Line Mechanism Obtained from the Locus Curves . . . . . . . . . . . . The Locii of the g-and C-Points for the Para- bolic Path . . . . . . . . . . . . . . . One Parabolic Path Mechanism . . . . . . . The Circular Path Problem . . . . . . . . . The Disc Rolling on a Disc. . . . . . . . . The Disc Rolling on Disc Mechanism for the case «=0 0 O O O O O O O O O O O O O O O The Rolling Curve Function Generator . . . vi Page 12 12 17 22 25 26 3h 35 39 #2 #5 51 Figure 18. 19. 20. 21. 22. 23. 2h. 25. 26. 27o 28. 29. LIST OF ILLUSTRATIONS -- Continued The Rolling Curve Inversion . . . . . . . . The Derived Path Tracing Mechanism . . . . . The Inversion for Constructing the Function Generator Mechanism . . . . . . . . . . . The Derived Path Generator Mechanism for the FunCtion $83 0 O O O O O O O O O O O O O The Function Generator Mechanism Derived for ¢=39L2 and Obtained from m1 at 0'0 . . . . The Performance Curve for the Mechanism of Figure 22 O O O O O O O O O O O O O O O 0 The Function Gengrator Mechanism for the Function .4180" Obtained from m; at a=O . The Performance Curve for the Mechanism of Figure 24 O O O 0 O O O O O O O O O O O O The Locii of the Points of Stationary Curvature and Their Centars of Curvature for the Equation #1 =3 9" . Equivalent Path Mechanism for v=3.hh50 9"aat ago 0 O I O O I O O C O O O O O O O O C O The Mechanism and the Performance Curve Based on the Solution of Figure 27 . . . . . . . The Figure Used in Deriving the Analytical Method of Determining the Output Angle . . vii Page 51 57 57 59 6O 6O 61 61 62 67 68 7O INTRODUCTION The four link mechanism. The four link mechanism is an assembly of four links pivoted together at their end points to form a closed chain. It has been studied exten- sively in the past and with good reason. It is the shmflest linkage device having constrained motion, does not require expensive machining to produce, and can be used in an end» less variety of applications. Such a mechanism can be used to produce plane motion or some input=output crank angular position, velocity, or acceleration relationship. The plane motion referred to here is the motion of the coupler bar, link b, in Figure l. The input-output motion is that of cranks a and c. For the linkage to be considered a mechanism, one link must be fixed.1 Other mechanism elements, such as rolling curves and cams, can be used to provide input-output relation- ships.293 They can be designed to meet exact position requirements over a given range, but ease of construction, lRolland T. Hinkle, Kinematics of Machines (2d ed.; Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1960) p. 7. 2Hinkle, p. 170. 3Alexander Cowie, Kinematics and Design of Megh: anisms (Scranton, Penna.: The International Textbook 00., I961) p. 368. l Fig. l.--- The four link mechanism. Fig. 2.» The crossed four link mechanism. 3 good wear properties, positive constraint and perhaps other virtues make the four link mechanism the most attractive when it can be designed to provide motion similar to that of a pair of rolling curves. The procedure begins with a pair of rolling curves, one being fixed and the other mov- ing, which are synthesized from the motion requirements. The writer has found no reference to a procedure in which a pair of rolling curves is synthesized and these curves used to develop an approximating four link mechanism. Only a few functional relationships between input and output cranks can be satisfied exactly by a four link mechanism.4 There are, however, many applications where a close approximation to a given functional relationship over a limited range is all that is desired. Such require-~ ments can usually be satisfied with a four—link mechanism. The design of a four link mechanism to satisfy some coupler bar or output crank motion for a complete cycle of the mechanism is beyond the scope of this work. Attention is directed toward the problem of devising mech- anisms which approximate a given required motion over a limited range. The problem can be divided into two cato egories: path generation and function generation. Path and Function Generation. Path generation is 4B. W. Shaffer and I. Cochin, "Synthesis of the Four Bar Mechanism when the Position of two Members is Pre- , scribed," Transactions of the ASME, v. 76, (Oct. l95h),P-ll37. A. the case in which some point in the plane of the coupler bar (link b of Figures land 2) traces a portion of some pre- scribed path. There are practical applications in machin- ing surfaces and providing special motions in machines.5’6 In the case of function generation, crank c has some par- ticular motion relationship to crank a. The prescribed motion may satisfy requirements for position, velocity, or acceleration. There are useful applications in the 7 fields of control mechanisms and computing devices. There are two basic approaches to both cases. The first, as applied to function generators, consists of choosing several values of the independent variable and computing the corresponding values of the dependent vari- able. A mechanism is then devised such that the output crank passes through several angular positions representing the dependent variable during the same phases in which the input crank is in angular positions corresponding to the independent variable values.8’9"lo 5James C. WOlford and Donald C. Haack, "Applying the Inflection Circle Concept," Transactions of the Fifth Conference on Mechanisms (Cleveland: The Penton PubIIsHIng Company, 1958) p. 232. 6Joseph S. Beggs, Mechanism (New York: McGraw Hill Book Company, Inc., 1955) p. 200. 7Hinkle, p. 293. 8Ferdinand Freudenstein and George N. Sandor,, "Synthesis of Path Generating Mechanisms by Means of a Programmed Digital Computer," American Society of Mech- anical Engineers Paper No. 58=A~85. 9Ferdinand Freudenstein, "Approximate Synthesis of Four Bar Mechanisms," Transactions of the ASME, v. 77, (August, 1955) p. 853. 10Hinkle, p. 267. 5 There are, mathematically at least, an unlimited number of possibilities in devising mechanisms to match a given mo- tion requirement. The second type of solution is based on the geome- try of the plane motion of the coupler bar and is usually 11,12 It is referred to as the inflection circle concept. based on the theory of roulettes, which treats of the paths of points in the planes of curves as they roll without 13 This theory can be used to slipping on other curves. develop the Euler-Savary Equation which is used in apply- ing the inflection circle concept to determine the center of curvature of the path of a point in a moving plane. In application, the inflection circle can be used to synthe- size mechanisms which have motions matching a given re- quirement over a finite range of displacement. There are. many examples in the literature.lh’15..As with the pre- cision point method, there are infinitely.many.mechanisms obtainable.from this method which will satisy a motion re- quirement over a small range. 11Allen S. Hall, Jr., "Inflection Circle and Polode Curvature. " WWW- agisms (Cleve and: The Penton Publishing Company, 1958);n207. 12Allen S. Hall, Jr., Kinematics and Linka e Desi n (Englewood Cliffs, N. J.: Prentice-HaII, Inc., I96Ii,p. 6%. 13Ben amin Williamson, An Elementary Treatise on the Different al Calculus, (London: Longmans Green and Co., LTD, 1927), p. 335. lHWolford and Haak, p. 233. 15Hall, p. 9b. 6 Synthesis methods using the inflection circle con- cept are not well suited to strictly analytical methods. Since the programmed digital computer can do numerical work with such rapidity, the use of such a device for mechanism synthesis seems very attractive. The deve10pment of such methods however, necessitated a reconsideration of the underlying mathematical theory. C HA PTER I THEORETIC AL DEVELOPMENT THEORETICAL DEVELOPMENT Roulettes The roulette. A curve generated by some point in- variably connected to a curve which rolls without slipping 16 Two well known examples on another curve is a roulette. are cycloids and trochoids. In Figure 3, the curves Cl and C3 are trochoids while curve 02 is a cycloid. If the co- ordinate system in Figure 3 is located so that the origin is at O and the radius of the circle is r, then the para- metric equations for the location of a point n, which lies in the moving plane, are: x": ra—G'nsine (I) Y" r -O-r3cose (2) Equations (1) and (2) can be used to devise a mech- anism having four rigid, hinged links, one link of which will closely approximate, over a small range, the motion of the circle when rolling without slipping on the straight line. To do this, one locates a pair of points in the plane of the circle which have "stationary" or unchanging l7 curvature. Since the general point must be moving 16Williamson, p. 335. 17Ha11, p. 97. AA ' /%? , Fig. 3.-- Three roulettes: The cycloid 02 and the trochoids C. and C3. Q # ' / Fig. 4.--A four link mechanism to replace rolling motion. 10 along a trochoidal path with respect to the straight line, the curvature at every point is changing more or less rape idly. The points having the greatest range of stationary curvature are best. After the two points are selected, their centers of curvature are determined. The straight line is considered to be a line scribed in a fixed plane, and the two points chosen in the moving plane are joined by means of rigid links to their centers of curvature lo- cated in the fixed plane. See Figure 4. This method of synthesis will apply to any pair of rolling curves. A more general expression than equations (1) and (2) will be required; one which will include the rolling curves used. Since the shape of the rolling curves is not generally known in the beginning, some method must be devised to determine the expressions of the curves as an intermediate step. This is done in the following consideration of plane motion. Plane Motion and the Rolling Curve Pair Plane motion. The plane motion of a plane may be some combination of translation and rotation. Whatever the motion, it may be considered to be composed of a num- ber of small rotations about different instantaneous cen- ters. Hence, any plane motion is the equivalent of the 11 rolling of one curve on another.l8’19 The given motion can be reduced to a rolling curve pair by using the concept of the instant center in the manner shown in the following development. Refer tc>Figure 5. A moving point P is located in a left-hand coordinate system x-y, which moves in a fixed right-hand system X—Y. The location of P in the X-Y system in terms of the parameter ¢, the angular displacement of x-y with re- spect to X-Y is: Xp= XQ-rxpcos¢-ypsh1¢ (3) Yp=Yo+xpsin¢+ypcos¢ <4) XQ and YQ locate the origin of the mOVing set in the fixed set. The location of P in the x-y system is made simi- larly (refer to Figure 6.): xP = x0 + XPcos¢ + YP sin4> (5) yP= yo - XPsin4> + YP cos 4. (6) XQ AND YQ are related to xO AND yo BY: l8Williamson, p. 363. 19Edwin Bidwell Wilson, Advanced Calculus (Boston: Ginn and Company, 1911), p. 73. Fig. 6.-- The location of the fixed coordinate system relative to the moving system. 13 x0 =~-(X0 cos¢+ YQsine) (7) yo s XQ sump - Yo case . (8) The fixed curve. The point P can be made to be the point of contact of the rolling curve pair. The requirement for pure rolling is satisfied if the velocity of the point of contact zero with respect to both systems: xp = 9, = KP = Yp = 0. Here the dot notation is used to represent deri- vatives with respect to time. The imposition of the con- dition for pure rolling at point P on the time derivatives of equations (3) and (4) results in: O = *0 - xpdasingb- ypducos e O = To + xpélcose — ypciasine which can be rearranged as: 1:9 = xp sin 4: + yp cos d: (9) £9 = ~(xp case -— yp sin 4:). ('0) The parametric equations for the fixed curve, which is in the fixed coordinate system, are obtained by substi— tuting equations (9) and (10) into equations (3) and (A). They are: Y X : x — q-Q (“Y P 0 ¢ YP = Ya + 5— (l2) ¢ 1h The Moving Curve. The equations of the moving curve are obtained in the following manner. Equations (7) and (8) are substituted into equations (5) and (6). The results are: xp = (XP- X0) case + (YP - Y0) sin ¢ 3n. = -(Xp-XQ) sine + (Yp - Yo)cos 96 Equations (11) and (12) are rearranged and substituted into the two equations above. The parametric expressions for the moving curve result. x... = $(XQ sin¢~ To c034.) (I3) yP = $020 cos¢+ 9Q sin¢) . (14) P is the point of contact between the rolling curves. It is expected that the path of Q will be expressed as some function of XQ. That is: v, = axe) Then equations (11), (12), (13) and (110 will be expressions written as functions of XQ and ¢. Obviously, it will be helpful if ¢ is also a function of XQ. For a particular path of point Q, Y will be expressed as a particular Q function of Xq. However, because the curve is a point path, the relationship between diand XQ is independent of the curve and any relationship may be used. Further, there will be a different rolling curve pair for every d»: g(XQ). This is not particularly important if the 15 path generator is to consist of a pair of rolling curves with the tracing point fixed in the moving curve, but if a four bar function generator with the tracing point fixed to the coupler bar is to be synthesized, the best function for ¢ will of necessity be determined by trial. This is so because the motion of the tracing point can be made to trace portions of some curves more accurately than others. A_Iglling_curve example. Some examples will be needed as the development proceeds so that principles can be illustrated. A simple one illustrating rolling curve development follows. The path to be generated is a straight line inclined upward to the right at an angle V/h with the following specifications for the plane motion: ya: x,, «in-xx, in which K is some assumed constant. From equations (11) and (12): x ‘x--—'- P- O K I HP 8 Yb'+ l<° This may be rewritten as: - 21 YP XP+K. The fixed curve is seen to be a straight line inclined up- ward to the right at the angle W/h from the X-axis. The line crosses the Y=axis at 2/K. The equations for the moving curve are: xP = ('IK)(sin4>- case) 16 yp = -(l/K)(sin¢ + cos 4:) After squaring both equations and adding, an equation of a circle about the origin having a radius of V27K is obtained. xgi-Y%= 27K - The tracing point is the center of a circle which will fol— low the line desired. See Figure 7. The Point Path Traced in the Fixed Plane Eqiations of the point. With the rolling curve pair expressed mathematically, the paths on the fixed plane traced by points fixed in the moving plane can be CD determined. Fmgn‘ 8 shows the rolling curve pair and the point path. Q is the origin of the moving coordinate set, and C is the moving point, which is located by: 62:67:».0‘; ((5) In order to write the X and Y components of equation (15), a radius vector angle a is defined. This angle, a position angle in the moving set, is measured counter- clockwise from the negative y-axis. The X and Y compo- nents of the position of g are written: x; XQ + 5—; sink-4:) (l6) Y; YQ + 52 cosh-4.) - (I?) The distance Q can be expressed as the product of some number m, to be determined, and the distance‘UP at the initial position of the rolling curve pair. 1? 0,0 x Fig. 7.-- The disc and straight. line rolling pair. Path of l; in X-Y Yi Fixed curve *0) chi \ + Q___1 - P Moving curve Fig.8.--The general tracing point and the rolling curve pair. 18 52 = mOP = High.“ 33:0 ((8) The expression under the radical in equation (18) can be represented by y. 7= =\/X2 Pit.“ YE) ,- 0 (l9) Equations (16) and (17) can now be written: X§= XQ + my sink-(p) (20) Y; 3 Yo + m7 C05(a"¢) o (2') Points having stationary curvature. The points in the moving plane which trace paths having momentarily sta- tionary curvature can be located by means of expressions for the radius of curvature. Radius of curvature express- ions can be obtained from any textbook on the Calculus.2O The procedure is to equate the derivative of the radius of curvature, with respect to the position angle a, to zero and solve the resulting equation for m. Equations (20) and (21) can then be used to locate the stationary curva- ture points. The centers of curvature are located by using eXpressions obtained from the Calculus. (See equations (25) and (26)). There is, of course, a solution for m for every chosen value of a . 20William Anthony Granville, Percy F. Smith, and William Raymond Longley, Elements of the Differential and Integral Calculus (Boston: Ginn and Company, 1941), p.152. 19 The eXpression for the radius of curvature, taken from the Calculus, is: 3 2 (22) d Y: dxf where .21.... “lg/5‘3}. ch dt 23 dt () and dX sz d dex ——c— “It—z d2? dt dt2 dt dt2 (24) dx; 925;)3 dt The expressions for the centers of curvature are:21 X§" 91: I +(g%ifz ox = (ix; sz4 (25) d dV' 2 - Y . 4.2) dX GK and CY are the X and Y coordinates of the center of curvature. Equations (20) and (21) are now put into equation (22). The derivatives are determined, using equations (23) and (24): (The dot notation is used for time derivatives.) 21Granville, Smith, and Longley, p. 157. 2O 23;: ‘314- my.) sink-4i) (2.” dxt X0 - myél cosh-e) Eva" Yaxa + M7(M sin (01- -¢) + Ncos(a- 46)] + "2,2;2 ‘28) >2)? d C [XQ- myl cos(a at”? M and N are defined as: The expression for the radius of curvature can now be written: {XS-Y2 0+ 2my [Yo sin(a- d») - X0 cos(a- ¢)] + m . 3/2 R 3 2 24’2} C x070 9020 + M7[M sink-e) + Ncos(a- 40] + mzyz 4.3 . (3|) The process of writing the derivative of R; with respect to a can be shortened by writing R; symbolically as: The derivative is then: I / 3’2 3 2 dA - 13 ng_ (5)3A da A da cTa' ' B?- dR;_ If da -0, then d d 33—5 = 2A3. (32) da do Since A= X3 + if + 2my) 4- N cosh-M] + m2 72 $3 21 then .16.: 2m7$[70cos(a-4>) + kqsin(¢-4>)] D. .g = my [M cosh-p) - N sink-4”] When values of A, B, 3'3, and 3'3 are put into equation (32) and simplified, a quadratic equation in m is obtained as follows: m2y2432[sqi2- z] + myHlsM-thZ) sine-4.) + (3N +2XQZ)cos(a-¢)] - 3&0}, 5(0- XQVQ) - 206+ 7%) =0 (33) in which M cosh-4:) - N sink-e) Yo cosh-4i) 4- X0 sink-4:) (34) Equation (33) completes the derivation of the equations locating the points in the moving plane having stationary curvature and their centers of curvature. The values of m obtained from a solution of equation (33) are put into equations (20), (21), (25), and (26). For every value of a chosen there are two values of m, and it fol- kmm that along any line drawn through point Q and making the angle a with the -y axis, there are two points having stationary curvature. With the points of stationary curvature, {I and g2 and their centers of curvature Cl and 02 located, a mech- anism can now be constructed which will cause the point Q to move along a path which is similar over a small range 22 “’Té‘lk \Path at I.“ Fig.~9.-- The rolling curve pair and the derived approximating four link mechanism. 23 to that produced by Q as a point in the moving curve. Points Cl and Cl become the joints of a link joining the fixed to the moving plane; points :2 and Cg form another link. Application. The example presented earlier for rolling curve formation can be used to illustrate meche anism synthesis by this method. Some arbitrarily assumed values which simplify the problem are: a; = ii, = x, = 0. Then M and N from equations (29) and (30) are: M = K49; N=-K«'p3 also, from equation (34): z=¢i Equation (33) becomes: Zmzyzé.4 + me$4fsin(a-¢) - cosh-44)] - 2K2$4=O. By equation (19): -\/_2‘_ 7 ‘ |(’ The constant K is a scale factor and the geometry of the mechanism will not be changed by any chosen value. Unity is the most convenient choice and the quadratic equation in m becomes: 2m2 +V%[sin(a-4>) - cos(a-¢)] - I = O. (A) Any value of ¢ can be chosen since the character of the motion of a disc on a straight line is independent of the position of the disc. When ¢=O, the solution of equation (A) becomes: 21+ m=‘%- [cosa-sinaii/TT-sinZa] (8) Since a different solution of (B) exists for each value of a chosen, there are, mathematically, infinitely many possible mechanisms for drawing this straight line. The locii of the stationary curvature points and their centers can be obtained by computing the positions of the points and their centers for a finite number of values of a between 0° and 360°. The equations needed are listed below. Q§=%[cosa-sinaiVl7—sin2a] (o) Xg=QC Sin“ (D) Y; =0; cosa (E) i_ I+X Y; - W- (F) .._ QUXc-st‘QC) G Y9 - (l+-Y§) ( ) a .2 ex: xgs Y[';§(Y)] (H) '2 CY= Y§+ -L%1 (I) The locii are shown plotted inifignrelfih The nu- merical results of this program on a digital computer are shown in Table l. A single mechanism is constructed by choosing a value of a which is measured clockwise from the negative y-axis. In this case, with 4>=O , the negative y-axis 25 The §-curve is the locus of all points in the x-y plane having stationary curvature with respect to the X-Y plane. The C-curve is the locus of the centers of curvature. C- curve // __._ p Fixed curve 7/ ./ i/l / l / \ / \ / Moving curve \ / ' \\ / / / / Desired path of Q \\ \ \ \ 0,0 X} Fig. IO.--The locus curves for the straight line mechanism. 26 / —" «f Moving curve' Fig.ll.-- A straight line mechanism obtained from the locus curves. 27 ewem use: :0 oescwucoo MOH.0 4MN.H mHm.4H: #00.: 000. 000.H: 000. n0m.: mm0m.d «00.0 000. m0m.0~: 000.: 005. 040.: has. 04m.: 000H.4 000.0H 0M0. HHO.m0: 000.: men. 000.: #N0. m00.: mmH0.4 0H¢.mm 0d0. 00m.¢0N: #N0.: N00. .NNO.: 00m. m¢0.: 00m0.m 000.00H 000. 0mm.h0am: 000.: hem. mm¢.: 040. 000.: m400.m H00.00H: 00m. 400.0m0a: H40.: 00m. 400.: H00.H 000.:. 000¢.m Hmm.mm: 00m. m0m.0Hm: 000.: Ham. Hea.: 400.H m00.: mmam.m 400.04: mH0. mm0.04H: 400.: 000.0 000.0 00N.H 000.: 4H4H.m 000.0m: Nm0. 00m.00: 400.: Hm~.: 0nd. 0Hm.H 0mm.: m000.m 0m0.0m: 0H0. H4m.00: m00.: 00¢.: 04m. 00N.H m00.: om0u.w hmH.Hm: 000. 000.00: 0N0.: 000.: 0mm. OHN.H 0H0.: mba0.m m0¢.0HH: 0N0. 00H.mma:, 4N0.: 000.: mms. H00.H Ndm.: 0m4¢.~ NOM.00 m00.H 0H0.00 HOH.H: N00.H: ~40. 000. dm¢.: m00N.N m0¢.NN m:m.H ¢m0.¢H HH¢.H: HHN.H: 0H0. 000. 5mm.: 0400.N 000.0 Non.a b00.m 540.H: h0m.ai m00. 004. 040.: m0H0.H 0MN.m MNN.N 000. M00.N: 0Hm.H: 040. 0mm. 0mH.: 0m40.d 000.N H00.m 0m¢.: m0m.¢: 00N.H: 000. 000.0 000.0 m00m.d 0:0.H hom.¢ mm0.i 045.5: eom.Hc «00. mam.i Ned. 000m.a 0mm.H ¢0H.0 #00.: HON.¢H: 000.H: 000. 00m.: :00. mHNN.H 000. 040.0 000.: m0¢.0m: 000.: 000. ham.: 0m¢. 0b¢0.H 0mm. mm0.0a 000.: 00¢.m0: 000.: New. N00.: MNO. mmm0. 040. 40m.mm ¢N0.: 0H0.NNN: mm0.: N00. 045.: 000. 0000. 000. 00¢.00H 000.: 000.H00m: 0m4.: .04m. 000.: 040. mmwm. 00m. 000.00H: H40.: mm0.¢m0a: 40m.: 00m. 000.: 000.H 004m. 00m. H0¢.Nm: 000.: H0¢.0Hm: H4H.: Nam. m00.: 40w.a mend. Nwo H00 Nxo on max Hex NAM HAM < .Emficmzoos mafia pnwwmupm esp mom meopcmo hams» mam manpm>u50 sew icofipmum mo mpcecanoo 0 use x one you mesam> umpmasnme .H mHLmB 28 0H0. 400.04: m00.: Hm0.04H: 000.0 H00.: 000.: HON.H 0NON.0 ~00. 400.0m: 400.: H04.00: and. Nmm.: 0m0.: 0Hm.H 000H.0 0H0. 000.0m: 000.: mOm.00: 000. 004.: 000.: 00N.H 0mm0.m 000. 0NN.Hm: 0N0.: 004.00: 00m. 000.: 0H0.: OHN.H 0000.0 0N0. HON.0HH: 0N0.: 000.0MH: 004. 000.: H4m.: H004H 0400.0 000.H 000.00 NOH.H: 040.00 N40. N00.H: 404.: 000. 0004.0 04m.H 00m.mm MH4.H: H00.MH 0H0. HHN.H: 0mm.: 000. 00mm.m 400.H 400.0 000.H: 000.m 000. 000.H: 040.: 004. 0000.0 000.0 000.0 000.N: m00. 040. OHM.H: 0mH.: 0mm. 0000.4 400.m N00.N m0m.4: H44.: 000. 00N.H: 000.0 H00.: mHHO.4 maw.4 440.H 400.0: 000.: 000. 4ON.H: 04H. mam.: 00mm.4 000 H00 Nxo on max HAN N00 am» < .Ewflcmnooe mafia psmflmepm exp pom mempceo 04039 was weapm>pSo 0umcowpmpm mo mproa Mo mpqmcoasoo 0 use x esp pom mmzflm> Uopwaspme .emscfipcem--a magma 29 corresponds to the positive Y=axis. A line is drawn through Q at angle q. The intersection of the line with the {-curve locates the two points C1 and C2. These are the joints connecting the cranks with the moving plane. Lines are now drawn from El and C2 through P to the point of intersection with the C-curve. This determines the length of the cranks and the location of the joints on the fixed plane. The construction is shown in Figure 10 and the mechanism is shown in Figure 11. The portion of the path of Q as a point on the coupler bar is shown to indicate the range of match with the desired curve. Practicalityig The utility of a curve tracing device may not be too obvious. One possible application is its use as a special mechanism to aid in machining sur- faces. There is another possibility in replacing gears and rolling curves to provide a particular motion. In the example given, the link which is formed by drawing a line from g1 to C2 has the same motion it would have if it were a line scribed on the disc. This suggests the possibility of replacing a rack and pinion with a four bar when the range of motion required is small. Angular acceleration of the disc. The specifica- tion that d>be constant does not make this a special case, for as long as XQ = YQ = K¢>, the rolling curve pair will have the same form and the mechanism will be the same. CHAPTER II APPLICATIONS TO PATH GENERATION II PATH GENERATION Qpen Curves Open Curve paths. We shall now consider the problem of synthesis of a four link mechanism having a tracing point Q which traces some portion of an open curve. The general procedure previously described applies. The prin- cipal difficulty lies in the selection of the best relation- ship between ¢>and Xq. There are no general rules for se- lecting the relationship so that trial and error will per- haps be necessary. Parabola. As a first step in the exploration of the method, a mechanism is to be designed to trace a por- tion of a parabolic curve. 2 YQ = 4X0 (35) The e‘to XQ relationship is assumed to be The following equations are derived from equations (35) and (36). x,= {5 (37) x0: iii (38) Yo: (‘p/fl‘ (39) t0: «ii/«a.- — “5513. (40) 31 32 The fixed and moving curves are expressed by° XP3 CD "' l/V; (4') NF 2"; + l (42) y. = sine - %§=—¢ (43) x, = - (c... + $37.4 (44) The expressions for the roulette are: x; = 4: + mysin(a-¢) (45) Y; = 2V7; + my cos(¢-¢) (46) in which y:\/.£;E a Before a mechanism can be designed it is necessary to select some values of 4: and d) . By choosing 4’ ‘4’ = l the following series of equations is obtained. 7=Wf; M=l; Nf-d = cosh-4)) - 0.5 sink-e) cosh-e) + sin (a-d) Equation (33) then becomes: 2m2 + _lt_'l_ 5 +sin2(a-¢)- seesaw-4») _ 7cos(a-4>i + 5 sine-(b) :0 V2- 400501-43) + 55in(a-¢) 4cos(a-¢) + 5 sin(°‘¢) which can be written: 2m2+%D-E=O (J) if ___ 5 + sin 2(a-4>) - 3 cos 20g») 400561—44) + 53in(¢-¢) ’ _ 7cos(a-¢)+ 5 sink-4.) E " 4,030.4) + 53in(¢-4>)' (K,L) The following list of equations can be used to make the computations needed to synthesize a mechanism which will trace a portion of the parabola of this example. It was the basis for a computer program used to make a more complete solution. 33 m=Zl-‘/2 -Di'\/DZ+IGE (M) X§= l +V2m sink-4:) (N) Y; = 2 + Jim casual-(la) (O) . _ l + V? m sink-4)) (P) YC - l - f2- m coda-4’) .. _ - |/2 + f2 m [sin (OI-(p) - '/2 cos(1-¢) + 2m] YC ' [l-x/Em cosh—<10]3 (Q) .2 g _. ngI-t\t ) (R) CX X; Y'i') CY”= it + £11;%t-. (S) Y; The Fortran program for this series of equations is in the appendix. The tabulated results are presented in Table 2.. The locus plot of the points and their centers are shown in Figure 12. The performance of one of the possible mechanisms is shown it) Figure 13. Closed Curves Closed curve paths. The circlgi The next problem is to explore the synthesis of a mechanism having a coupler bar point which moves in a circular path. It is not to be expected that the resulting mechanism will generate a circle, but that it will approximate the curve over a lim- ited range. The problem is depicted in Figure 14. The equation for the path of Q is: (too 34 / / order of the branches. The arrows and letters on the C~curve are intended to show the C-curve »d P toe E-curve l I‘ | // O C-curve \ X ’- 0/ / / C-curve \a Fig. l2.-- The locii of the C parabolic path. and 0- points for the 35 0\’ <2... ‘ Traced curve Fig. l3.--One parabolic path mechanism. 36 fl (..:.,‘§10Iq I‘D’l r l! ’SUE- mil”. .IITII. ME:E. 004.0 000.0 404.0 040.4 004. 0 400.4: 400. 000. 000.4 000.0 000.4 004.0 000.4 040. 0 000.4: 000. 000. 004. 4 000.0 400.0 400.0 440. 000. 0 044.0: 004. 404.4 040. 4 000.0 004.0 000.0 440. 400. 0 004.0: 400. 440.4 000.0 000.0 004.4 004.0 000. 000. 0 000.4: 000.: 000.0 400.0 000.0 400. 004.0 0 4. 000. 4 000.4: 000.0: 404.0 004.0 040.0 000.: 400.0 0 0.: 400.0 000. 000.0 004. 040.0 040.4 004.4: 000.0 040.: 000.0: 000.0 400.4 404.4 444.0 000.04 004.04 400.0 000.: 000.00 044.00: 000.4 400.4 000.0 004.0 004.0 000.0 1004.: 444.44 040.44: 000.4 400.4 000.0 004.0 000.0 000.0 004.: 004.0 000.0: 000.4 000.4 040.0 000.0 040.0 040.4 004.: 000.0 000.0: 000.0 040.4 044.0 000.0 000.4 000.4 000.: 000.0 000.0: 000.0 000.4 000. 0 000. 0 000. 000.4 400.: 444.0 004.0: 000.0 040.4 400. 0 000. 0 004. 004.4 004.: 004.4 404.0: 440.0 440.4 040. 4 000. 0 404.: 400. 040. 400.0 000.4: 400.0 040.4 . 040. 000.0 000.: 000. 040. 004.0 040.4: 400.0 040.4 000.4 400.0 004.4: 040. 004. 000.0 000.0: 000.0 044.4 000.4 004.0 000.4: 400. 000. 004.0 004.0 004.0 040.4 400.4 040.0 000.4: 000. 000. 000.0 000.4 004.0 000.4 040.4 000. 0 044.0: 004. 004.4 000.0 000.0 000.0 040. 000. 000. 0 004.0: 000. 040.4 000.0 004.0 000.0 440. 000. 000. 0 400.4: 000.: 000.0 000.0 004.4 - 004.0 400. 000. 000. 4 000.4: 000.0: 400.0 000.0 000. 404.0 404. 040. 000.0 000. 400.0 404. 000.0 000.: 000.0 040.: 404. 1%):0004) 08 408 4.; Ex 4 .5000000 2400 044000000 on» non 004:000 .0 04000 37 400.0: 000.0 400.4 004.4 400.4 044.4: 000.0 040.: 000.0 000.00 000.00: 000.4 000.4 000.04 400.04 000.0 400.:. 004.0 000.44 000.44: 000.4 400.4 000.0 004.0 000.0 004.: 000.0 004.0 040.0: 000.4 000.4 004.0 000.0 000.0 004.: 000.0 000.0 000.0: 000.0 040.4 000.0 440.0 440.4 004.: 400.0 000.0 000.0: 400.0 000.4 000.0 000.4 000.4 400.: 004.0 004.0 004.0: 000.0 040.4 000.0 000. 000.4 400.: 000.0 004.4 004.0: 440.0 440.4 000.0 004. 004.4 004.: 000.0 000.0 000.4: 000.0 040.4 000.0 004.: 400. 040. 000.4 004.0 440.4: 400.0 040.4 000.0 000.: 000. 040. 440.4 400.0 400.0: 000.0 044.4 000.0 404.4: 040. 404. 000.4 000 000 00» 000 400 400 400 400 4 .8000000 0000 044000000 000 00% 0049000 .005040000::0 04000 38 (XQ-a)2 + (Ya-b)2 = o2 (48) in which C is the circular path radius. The angle W'is measured counterclockwise from the X-axis to the radius of the circle locating point Q, and is expressed by: YQ-b X a (4 9) a .. temp = The angular displacement of the moving coordinate system 18: ¢ = fHI) . (50) The equations for XQ and YQ are written by inspection of Figure la. X0: 0 + Ccos 4’ (5i) YQ=b + Csin‘l’. (52) Their time derivatives are: x0: - C‘itsimp (53) 70 = 04: cos :p (54) 20 = -C(:'p simp + ((30051?) (55) yo = C({II cosv/ -:ilzsinw) . (56) The parametric equations for the fixed curve are: XP=a +C(I- .)cos:p (57) ¢=b+cn- wa, 00 and for the moving curve: xp = - chit/<1» cosN-‘fi (59) yP = -C(:i:/$)sin(‘P-4>) . (60) Equations (20), (21), (25), and (26) are used to 39 Y? A 09- XV Fig. I4.-- The circular path problem. 1.0 locate the joints of the approximating mechanism. Equations (29), (30), and (34) are written as follows: M anti-4:46:00.» (Visupqiaicosw (6|) N = - C((ifié- Whoosw + (4)204. M2) sin :pl (62) 2: 0H:- :inflsinhpm ¢)+ (:i: Z¢+¢¢§cos(¢+a- 4:) (63) Vcoswm'fl The value of 7 is: 7:1/ xPz+sz =Cg. (64) After putting equations (61), (62), (63), and (6h) into equation (33). the result, after some simplifying, is: +.r_r12_[(<1>ii’f"“2 +¢¢2)sin 20- OH“ -W¢l(5+c0320)] (24:02- ¢:i:2)cos8 - (ipq'b :infi)sin0 -. 2[i¢,¢2_2{k2¢)c030 + (VJ: -:[::'f)sin8 ] = O (65) (2:):43- :thhose - (:‘H: - 4:4: )sinO in WhiCh 03“]!4-0 _¢). The four link mechanism which is to approximate this motion will be designed so that the tracing point Q fits at one point and values of 0, 0, (1;, :p, :i:, and :1; will be put into equation (65). This will simplify the equation. The coupler bar (attached to the moving plane) hinge joints are located at (Xglb Ycl), (1‘2, YCZ) and the hinge joints on the fixed plane are located at (0X1, 0Y1) and (0X2, 0Y2). The points and the derivatives of the path curves are: a + C[cos:4: + m(¢:/4>)sin(a-¢)] (66) b + C[sin:p + m(:i:/:)>)cos(x -¢)] (67) X; Y; #1 Y' = _ 4': c031: 4: mu! sinks-4:) +6 cosh-49)] (68) C 75in w + m [:l: cos(a ~45) ~£ small-96)] where £= flizté (69) ": .. fl! +M[Pcose +Tsin01+ Amz} Y; Cwsinw + m [:Pcos(a~¢)-€sin(a_¢n}3 (70) u_.... .2.. ”_2 where P = (W *9?+(’ 1') _ Hz— (7]) 4’ 95 T : "_ v . . - 2‘ . .. i and $04» w): 04..., $36) 4%,. 4,, (:2: A = *‘i’w’a’. ’3WP‘4’W) + M¢$4+2$W + 30,392... $0102 (73) :3 - The centers of curvature are determined by put- ting equations (66), (67), (68), and (70) into equations (25) and (26). Circles rollinggon circles. The foregoing pro- cedure may be used to synthesize mechanisms which approx- imate the motion of circles rolling on circles. In this case, the fixed center 0 is shifted to the center of the circular path. See Figure 15. Here, _ D+d 0" 2 wie- (7 4) The moving circle motion can begin at YQ = O, XQ = C. After an interval, 55 rotates an angle W and the moving circle rotates through angle x with respect to the line 56. The moving system, attached to the moving circle, turns through the angle X‘+4I: 1:2 \ ° \ .Q. 2 Fig. l5.-—- The disc rolling on a disc. A 1:3 4, -.- A + :p. (75) From Figure 11.: id = (:0 (76) and 95 ‘P on - (77) By substituting n= d , 4:: my , $3“- m): , and (i3: nit? into equation (65), the resulting quadratic in m becomes: I . m2+ Zmé‘fi—‘sme - 2n—l = O (78) in which ®= 4:-4>+a 7_ C%- Dzn+d The points of stationary curvature are: x; = -";"[cos:(: + 1} sin(a-— qb)] (79) YC = 92-‘1[sin :p + -';','—cos(a-4>)] . (80) Expressions (69), (71). (72), and (73) become: 5“ O P _.; JV—JI-nfi .. 'r = :pmrm) ~2¥¢31+§ Wrflifi- 2nd::1)2+ 2n2 2 2 . n3q/3 The geometry of such a mechanism would not be A altered by taking é:= I, :‘p=’:1:'=o , so that P=O T=n+l A=n and the derivatives can be written: . cos :p + m sin(a 1:) = - sin q: + m cosh—4:) (8') 4h 2 I +m(n+l)sino JI-nm2 D+d [s:n::+mcos(a-¢)]3" Y; = ' (82) Application. A mechanism is designed by putting values into equations (78), (79), (80), (81), and (82). As an example, for d = 2, n = 1:», 4" 1r/2, ¢= 2-1r and equation (78) becomes: 2 2. _.EL m+7mcosa 730 for <1= 90°, ml = 0.535, m2 = -O.535. The coordinates of the joints are: x§1= 0.535; Y§l= 4.0; cxl = -o.7hh; 0Y1 = 1.608 x£2= -o.535; Yt2= 4.0; 0x2 = 0.7hh; CY2 = 1.608 The mechanism is constructed in Figure 16. fieduction of the case of the disc rolling_on a disc to that of a disc rolling on a straight line. In the extension of this case to that of the disc rolling on a straight line, the diameter D becomes infinitely large. As I D—v-oo; n—u-oo; s=-n—->-O , Then, replacing n by l/s, equation (78) becomes: 2 3+) . _ 1'25 g m + [—2_s]m smO 2-s O. (83) By putting s = 0, equation (82) becomes: m2+ flsine - -'— =0. (84) 2 2 Now, by using the trigonometric identity: sin®= sin(:[:+a -¢) = sin:.(:cos(a-4>) + CostiMa‘cfi) , 1+5 92 5| Path of Q '°" 0 ‘ Moving Polode F' d Ixed polie P Cl C2 _> O X Fig. l6.-- The disc rolling on disc mechanism for the case a=0. #6 and taking, for a horizontal line below the disc, the value v = "/2. sinO = cosh-4:) and equation (82) becomes: m2 + Ian-cosW-M-‘é'; 0. (85) Now, if the value for W is taken as -774 in the first case considered, in which the disc rolls on a stranyn; line above the disc and inclined upward to the right, sine = -gcos(a-¢) + £- sin(a -¢) 2 Equation (82) becomes: m2+ 2135 [sin(a-¢)-cos(a-¢)] - i = 0 (86) which, when multiplied by 2 is identical to equation (A) of the first example. Thus, the development of the mech- anism to generate a portion of a straight line path is a special case of the more general curved path case. It is to be noted that the mechanism develOpment begins with a different equation for m for each differently inclined line, and for each line there is a very large number of possible mechanisms. CHAPTER III FUNCTION GENERATION III FUNCTION GENERATION Function Generators. Four link function generators. The relation be- tween the input and output link angular displacements of a mechanism is a geometrical property. A mechanism so de— signed that the input-output relationship satisfies a par- ticular mathematical function between two quantities is a function generator. Function generators.may be used as components in control systems, instruments, or as mechani- cal analog computing elements. Mechanisms which match any given function relation- ship exactly can be constructed from rolling curves.22’23 Such devices can be difficult to machine and when the input output requirements are not exact or a small range of motion is required, a four bar function generator may suffice.) For the purpose of this study, the four bar function generator is a mechanism constructed of bars or links, the lengths cfwhich are such that the crank angles oorrespcnd to the variation of some dependent f ‘Vfi 22H. E. Golber "Rollcurve Gears," Transactions of the ASME, v. 61, (1939) p. 223. * 23Begg8. p- 7h- #8 #9 variable and the variation of its independent variable. The devices considered here will be approximate function generators. Function generators can be synthesized by seeking a mechanism such that the output link will be in certain positions when the input link is in correSponding positions, the positions being obtained from several numerical solu- tions of the desired functional relationshiplzh’ZS There are an infinite number of possibilities mathematically, and a very large number of different mechanisms can be obtained from one set of precision points. The mechanism positions between precision points are in error and part of the problem is that of locating the precision points so as to 26 minimize the error. Mechanism synthesis based on the inflection circle concept is a different method.27 As pointed cut earlier, that method and the method presented here are based on the same fundamental theory, which is one of matching the mech- anism performance to the function over a small range. The application of the roulette method. The direct 2(“Ferdinand Freudenstein, "Approximate Synthesis of Four-Bar Linkages," Trans. ASME, v. 77, (Aug., 1955), p. 853. 25Hinkle, p. 267. 26Ferdinand Freudenstein, "Structural Error Analysis in Plane Kinanatic Synthesis," Trans. ASME, v.81, ser.B, n.1, (Feb.l959 ), p.15. 27Hall, p. 106. 50 application of the method of roulettes to function genera- ticn requires beginning with a rolling curve pair. The curves wfll be expressed in parametric fcrm as discussed previously. The procedure for designing rolling curves is well known.28 Any pair of rolling curve function generators which can be designed by Golber's method can be used as a basis for four bar function generator design. The procedure for synthesizing a circular path generating mechanism is combined with an inversion of the rolling curve mechanism about one curve, preferably the input curve. The rolling curve function generator is depicted in Figure 1?. Angle 9 is the input. Figure 18 shows the in- version of the mechanism about the input link and is the basis of this development. Since the rolling curve mech- anism is usually designed with fixed centers, the path of the ground joint of the output link is a circle. The output angle, W , when added to the input angle, becomes ¢, the displacement angle of the moving coordinate system. The case of the circular path generating mechanism with the circle center at the fixed origin applies. The func» tion generating mechanism resulting from this synthesis, however, is quite different from the path generating mechanism, as shall be seen. The moving system displace- ment angle is: 28Golber 51 Fig. l7. --The rolling carve function generator. Fig. IS. --The rolling curve inversion. 52 ¢=9+.p. (87) Rolling_curve design. It is to be assumed that some function is to be generated by a pair of rolling curves such as 0: Ma) (88) and that the curves have a fixed center distance so that R+r= C (89) R and r are associated with input and output angles, 0 and irrespectively. The condition for pure rolling is satis- fied by: RdO = rd‘p. (90) By rearranging equation (90) .8. 3 £311 9| r d0 ( ) so that _ i r R dw/da (92) and finally: C(dwda) (av/d9) + I (93) and r = I (94) C (deBh-l ° Equations (93) and (9h) eXpress R and r as func- tions of the derivative of the desired function, which is in turn a function of the independent variable, 9. Rolling curves using instant centers. The rolling curve development by the method of this thesis can be com- pared with the method of Golber referred to earlier. 53 The path of Q is now taken to be a circle about the fixed origin and its equation is: X3 + Y::= 02 Here, C = OQ. Since a .tafiilh x0 then X0 ‘0 cos 9 Yb==05h19 . The time derivatives are; *0 '3 -Césin9 Y0 = cacos 0 . The parametric equations for the rolling curve pair are (95) (96) (97) (98) (99) (IOO) obtained by putting equations (97), (98), (99), and (100) into equations (11), (12), (13), and (1h). Xp= cu Air/6)]... a VP = on ~(é/imlsin 9 X9 = -C(é/$) cos(9-¢) YP '-‘ C(é/d) sin(9-¢) . (I0!) (I02) (I03) (IO4) Comparison of results with Golber'§=method. To com- pare this with Golber's method, it is noted that: R=VX§+Y§ and r=Vx§+y§. Then emu-(wan r = C(84) . Replacing 4> with VH8: “05) ((06) (I07) 54 (W/0)+ l C = . . (H39) and (W/9)+’( since @Vé= fig, it is seen that equations (108) and (109) are the same as equations (93) and (96). Synthesis of the Four Link Function Generator. Basic equations; With a'b=0 in equations (65)amd(66) and with w replaced by 9, the equations of the point on the roulette are: x; = ()[0059 + m(é/qi>)sin(a-¢)] (n0) Y; = C[ sine + m(8/$)cos(a —¢)]. (III) The location of the points of stationary curv- ature is made as before, that is, by solving equation (65) for m and putting the values of m into equations (110) and (111). Equation (65) can be simplified by noting that the geometry of the four link mechanism will not be changed if the input link has constant velocity, that is, m2[&>(2&>-é)cos® + $sin 01+ 3%) [JQHQ sin 20 +$(5+cos 20)] - [8($-28)cos® +$sin®] = 0. (”2) In this equation, 9 =9-¢+a . The mechanism is constructed to fit the desired 55 function exactly at one point. This point can be selected in the center of the desired Operating range with the ex- pectation that good approximation will extend an equal amount on either side of the matching point. Whether or not the mechanism will satisfy the requirement over the range desired cannot be determined since the range of best fit cannot be found at this time. When the matching point of the mechanism is chosen, values of 9, 8, 6, (ii, :3, and 4; Will be fixed, This will simplify equation (112). The points of stationary curve ature are determined by inserting values of m from eq- uation (112) into equations (110) and (111). It must be remembered that a pair of values of m are obtained from equation (112) by fixing all variables, so that the same variables must be used in any given solution of equations (110) and (111). Equations (25) and (26) are to be used for the centers of curvature, using the following deriva- tives: . - _ ' '2 - Yc' : __ cosB + m[sm(a 4:) (if/4: )cos(a 4:)1 (IIS) sin a + m[ cosh-4:) + ($l62)sin (cl-49)] 8+mRW) sinO - Qigicos 8) + $2 R¢2+9$'2$2)' $2] Y" =- ' C .66 '9 [ (wine->13 {sun +m costlcti-lr‘fi2 sun 4> } (II4) where ®=9-¢+a. Function generator linkage construction. The den 56 rived mechanism for path generation is illustrated in Figure 19. The point Q is intended to describe a segment of a circle. Cl and :2 are the points of stationary curvature and points 01 and 02 are their centers of curvature. This is the inversion of the function generator. It is to be noted that in the origional motion, the XQY system is to rotate about the point 0 through 9 while the x-y system is to rotate about the fixed point Q. The distance 05 then is required to be fixed and to be made so by using a link which becomes the fixed link of the mechanism. Next a link is used to attach points C1 to C1 and the two planes, XeY and x-y have constrained motion with respect to each other. These two planes can be reduced to links 051 and 5:1 and the result is the four link function generator. A second possibility is mechanism 002920. See Figure 20. Application. An easily followed procedure for numerical synthesis can be devised by putting the so- lution of equation(112) in a more easily handled form. If the mechanism is a position function generator, i. e., velocity of the input link is not specified, then 8= 1 can be inserted into the solution. The result is: m=-,-:-(-01'\/02+I6I-:) (us) where 3 j>($+l)sin20 +§(5+cos28) (ll6) 6(28-Iicos0 + $sin® D 57 Y“ 0| O/ 0 f Fig. I9.--The derived path tracing mechanism. Fig. 20.--The inversion for constructing the function generator mechanism. 58 and 6($+2)cose -$sine . II7) $(2¢- I) cosO + {fisino ( E: Example. An arbitrary function is chosen to illus- trate the procedure. |P=:30L2 This is put into the form to be used in this synthesis method by replacing (it by (fr-9: (This accomplishes the in- version) ¢= 39"21-8 with time derivatives: $=(3.69°2+ mi 3;:- (0.72 9‘3) 02 '4; =(- 0.576 9'”) 63 For 9:1, 0:1, 4>=4. 6:4.6, 6:072, 32-0576. (“0) = -().3|S), IE= O.3|4 [0.319 £40319? + l6(0.3|4) J J. 4 = -0.688. The following values "‘3 so that ml = 0.6h2, m2 are obtained: X§l= 0.6h5, Y;l= 0.736, CXl= 0.065, CYl= 0.505 XC2= 0.460, Ytz= 0.907, CX2= 0.358, CY2= 0.173. The constant C is a scale factor and can be taken as unity. The derived mechanism is shown in Ennre2l. The function generators resulting from the inversion are shown in Figures 22 and 24. The performance curves, graphically determined, are shown hiifigures 23and 25. The locus plot from a com- puter solution is shown in Figure 26. 59 / / l l 9 02 I / /// Fig. 2|.-- The derived path generator mechanism for the function u: = 30" 60 Fig. 22.-- The function generator mechanism derived for \P =39"2 and obtained from m. at a=0. / / Output angle 4:, radians .f 0.6 0.7 0.8 0.9 1.0 H 1.2 (.3 (.4 -———r—-Desired curve Input 9, radians Obtained curve Fig. 23.--The performance curve for the mechanism of Fig.22. 61 Fig. 24.-- The function generator linkage for w 3:59"2 obtained from m2 at a: 0. 4.0 ::' . .§ ,0 _. .E at. 3. .. 3.0 '3 9- E 3 / as v’ / g 2.0 /’ ‘5 // .9 ,/ 3 0 IO 0.6 0.7 0.8 0.9 (.0 LI |.2 (.3 L4 --— Desrred curve Input 9’ Radians Obtained curve Fig.25.--The performance curve for the mechanism of Fig.24. 62 C2 d to e \ Q {-curve \ CI b to c I c —- "I' I C-curve } The letters indicate the curve order. A mechanism is constructed (:2 by drawing a line through 0 I’ to locate g, and g2. C, and Cadre located by drawing lines from g, and ;2 through P to the C-curve. - . y... ftoa Fig. 26.-- The locii of the points of stationary curva- ture and their centers of curvature for the equation ip=3 9L2 63 mom. men. one. mos. med. sea. mam. 6mm. Nom.s mum. woo. mmo. 00m. 00H. wea. mom. Ham. mma.¢ 5mm. Hmm. umo. 5mm. mma. sow. mom. Hem. mao.4 Nmo. mum. 0mm. Hum. mmfl. 0mm. Ham. mam. omm.m amp. and. one. 0mm. mma. mam. cam. sam. emm.m awn. woo. now. Now. Hma. mmm. mam. was. omd.m mam. duo. was. umo. mma. «mm. mam. owe. mHm.m Hem. omo. 0mm. new. 00H. 0mm. 0H0. one. H4H.m ems. Ono. cum. mew. sow. mom. moo. mmq. omo.m was. mmo. cow. new. man. oee. mmm. was. mms.~ awn. one. Ham. new. man. one. mom. mom. bam.~ Ham. one. 04m. one. 00m. saw. mew. hum. m¢4.m 6mm. smo. «mm. amp. Hum. can. mam. mom. mmm.m com. 00H. Ohm. «No. n04. «mm. mun. com. doo.~ How. sad. mam. mac. mos. man. man. 40m. mam.a «mm. mma. mom. moo. NOm. mom. won. mum. men.a com. flea. mmo. map. emu. sum. mum. mom. oem.a Hus.a oae.i cam. mum. mmo. H04. 4mm. wme. mom.a mud. mad. mmm. own. man. mom. one. mes. HNN.H oma. mua. mom. Hum. men. can. mmo. 00m. 540. a mmH. 40m. mow. Hem. 0mm. 0mm. 5mm. 5mm. mum. «ma. 0mm. Ado. mmm. «no. mum. Ono. Hum. woo. «ma. mam. ado. 4am. mmo. end. one. 0mm. mum. awn. mmm. cam. was. eon. moo. map. «No. man. mma. mum. mam. owe. men. duo. Nae. ems. sea. «Mo NNo mm» NAN HMO on an» HAN < .Emswona uopmsmCmm :ofipocsm esp pom muasmmm .m magma 6h 06H. mam. cam. mas. Hem. one. One. «so. mmm.m sou. man. was. ems. was. Ono. see. mss. nos.o cam. oss. mmm. was. was. «no. com. ess. mmm.m mum. mes. mom. mom. awn. one. saw. mss. mme.m com. sen. msm. sen. Ham. oeo. osm. ems. sum.“ Hum. osm. mam. «on. own. smo. 6mm. mus. mos.m eos. mm“. wee. com. com. cos. one. mms. mm~.m mos. on“. mse. son. How. was. «mm. “as. coo.m Now. non. mos. sum. «mm. was. - mom. mos. 6mm.s Mme. mum. mes. mam. cam. and. mum. was. Hae.s mmo. Hos. smo. oms. How.i ems.u Ham. mum. smm.s «so mxo New mix awe axe as» sex s .smpwopm sopmuocow cowpoczm esp you madammm .nmscfipcqmrnm magma 65 The mechanism is expected to represent the function to be generated over only a small range of values. It will be necessary to decide on the range of values of the func- tion the mechanism is to generate. In the example just given the scale divisions of the function and the angular diSplacement of the cranks in radians corresponded. In another case it might be desirable to expand or to com- press the function scale. In the absence of any specific information on the mechanism, a limitation of something less than 180° rotation of the output link can be imposed when the range of values of the function to be generated is greater than that of its independent variable. This particular mechanism will probably be a crank and rocker mechanism which is driven by the rocker. To expand the range of the function to be generated, in this case, a revision of the equation relating the in- put crank angle to the output crank angle is necessary. By rewriting the function, i. e., renaming the variables; y=E§xL2 The relationship between the input angle range and the independent variable range is: A6=KAx For the two variables to have equal values, K equals unity. For A9=I, and Ax=2, K= l/2 then x=29, y=2tp and the function to be generated can be determined by substitution into the function statement: 66 (24’) = 3(29)L2 or 4. = 3.445 0"2. After replacing (p by (p-G and solving for 6: a = 3.4456“?- + a J. =(4.l49'2 + Ilé a; =(0.8289"°)é2 “a; =(-0.662 9"”)63 for x = 1, e= 0.5, é= 1, 4.: 2.0, ¢= 4.6, $=1.z.z.3, 4> = ~1.757. Choosing a= 0, the calculated values are: D = «0.875, E = 0.10, ml= 0.887, = -0.151. Then: m 2 x51: 09365, Y§2= 0.762, CXl= 0.916, CY1= -0.l+(,0 x§2= 0.629, Q2: 0.883, 0x2= 002,0, 0Y2. 0oh67. The stationary curvature points and their cen- ters are plotted inFigure 27. The derived mechanism for path generation is shown. The function generating mechanism, constructed using points 0, Cl, :1, and Q and inversion about link 001 is showr in Figure 28. Note, however, that the input range has only been extended a small amount and that the degree of fit is not as good as that of the first mechanism constructed. (Fqgue 24‘ However, there is probably still a better mech- anism and it can be obtained by taking another value of a to be put into the equations. A locus plot is an interesting study to make. 67 ('5 at «=0 =3.4459 t path mechanism forg, len -- Equiva Fig. 27 68 L5 :2 y 0 fig, 1: 5.0 / // 4.0 CI /// / 3.0 e. / $2.0 / a x 7 f ——- Desired curve. / —— Obtained curve. Lo/ 00.5 0.6 0.7 0.8 0.9 |.O Li (.0 (.3 L4 Input,x Fig.28.--The mechanism and the performance curve based on the solution of Fig. 27. 69 The Analytical Determination of the Output Crank Angle for the Four Bar Function Generator. With the avail- ability of rapid computational equipment, the determina- tion of the output angle of the derived four bar linkage by numerical rather than graphical methods is desirable. In addition to the speed there is the availability of greater precision afforded by numerical methods. Figure 23 shows the pair of derived mechanisms which are obtained from a solution using one value of (I. An expression relating W and 0 is to be obtained. The angles of the inverted mechanisms which correspond to the input angle 9 of the function generator are n and v . The links are: (The subscripts are omitted.) a = 00 = V(cx)2 + (CY)2 (“8) b = c; = “(X -cx>2- (Y -cv)é (”9) c -"-’ QC -‘-' my (I20) d = 00 d is a scale factor. Unity is a convenient value. TWO relations for the output angle w result. They are based on the following: 1")0 p>1r Case I P46 [1(11' Case 11 also: - ~19. ~ F-—tan cx UZl) 7O ’- X Fig. 29.-- The figure used in deriving the analytical method of determining the output angle. 71 Case I. For case I, 1; = l"-9 , ((22) The line 3 = 0611s drawn and by the use of the law of co- sines: s2: 02 + d2 -20d cos 1) (l23) The two angles B and A are defined from: B = sin"('% sin a) (124i _ 2 x=cos.'2(sz+sc: b). (l25) The two expressions obtained from Figure 29 are: P = 1523-49-4. +a) p=w+WX-B) from which: 4>=9+a -g-+().-,3)0 (l26) By substituting ¢=9+W9 4"“'-2'L*("‘B)° ((27) Case II. For case II, n==9-P “21» and the expressions for F are: [1,: %+(9-¢+O) p1: 7'f(k"B) from which - —- — ¢-9+a+2 (x B) and finally: \I/= a+%-(X-B)- 72 The computation is carried out in tabular form in Table 4” The output scale is to be regarded as movable, and the position is to be determined by the matching point. In this case, the match is made at 9= 1 and qr= 3. For this reason, the equations for p and W are not numbered. In computing the output values it is only necessary to de- termine (x -B) and the difference between the required angle at 9 = 1 (¢= 3), subtract (1 -3) from this angle and add the difference to (x -,8) for the other values. 73 eNH.- oma.- ems.- mos.- moo.- pso.- Hoo. moo. own. mos. sow. ms . N. + N. 3.... E. op 23 as: in ms... tom o5... 6.8 pom e omo.- Nao. was. oom. How. mom. oos. oss. oes. sos. son. p new m\e emo.- «so. mas. Ham. mom. oos. Hos. pom. “no. ewe. Hoe. s.:sm omo.s mmo.d moo.d ooo. “so. moo. mmo. Hoe. man. oop. emp.e .\8 sms. Hos. sms. man. man. com. oap. mmp. non. mso. omo. m ssm. ssm. ssm. mom. mom. mum. mum. sms. oom. own. «mp. mm “Ho.s oso.s mso.a poo. oeo. smo. sow. smo. one. ooe. sup. 1.866 new mom. mom. moo. mom. «no. paw. com. mam. mso. pmp. map. .9 moo o..- no so «.2 0.3 6.8 saw 13 mos ops em. 2...: L. Nag opp Now use m.mp imp QR pém m.ms ads s.sm Two“; o p.s m.a s.s m.s N.H H.H o.H o. o. -u. p. l.ommo o 6.3 . Wmmwm 7:3 a o .oooA .. 8 £36 .. o capo .. 6 .oomo ._ m .o a . .sm osswfim mo popmuoCmm cowpocsm on» non pzopso poundEoo .s eases 74 8.. $.s 33 :s RA RA ooémsos .93“ no; 83...; ops pp.s oss 33 EA sm.m oo.m 01:..N mom.m mmé i 756.: 4. wow epm emu mam saw .oos mes ems and mod -- 1.8881 s oS was SH posing toe s.~p Naps «.8 mt? .... of; sad oms smH mmH NNH p.so p.pm s.mm m.om m.o~ i- p:.2 moo... no... mom... 3... R... om... so. pom. so. .3. mi; 2 .8 osH. oms. oss. psH. mas. mos. mes. mos. oom. mam. pom. omm p.s m.H s.H m.s ~.H H.H o.H o. m. e. p. l.oamo sopmsocom coopocsm esp sow pseudo oopsmsoo .sm assess o6 Apozcflpcoov .s magma IV CLOSURE CLOSURE 0n the preceding pages there has been no attempt to derive any existing equations. The origional intent was the development of a method for optimizing a mechanism synthesized by the inflection circle concept. It was first determined that some expressions of the procedure in an analytical form were necessary. Examination of an out of date calculus book uncovered the theory of roulettes and its use in developing the inflection circle and in deriving the Euler-Savary equation. It seemed to be the necessary theory on which to base an analytical synthesis method. This development was the result. The basis of this work is the plane motion of a plane: every mechanism synthesis here begins with instantaneous plane motion. The plane motion is described in terms of a pair of rolling curves as an intermediate step in the location of a pair of points which are moving in paths having instantaneously stationary curvature. If the radius of curvature of the path is constant over a considerable displacement of the plane, the coupler bar plane of a linkage having joints located at these points will match the origional plane motion for the same range. Also, if the curvature is not changing 76 77 rapidly there will be a good match. A rapidly changing radius of curvature means a poorly matching mechanism. In the event of a poor match there are two possibilities: the first and easiest is to select a different position angle, ¢, The second possibility is to select a different plane position. In the second possibility it will be necessary to rewrite all the equations. In either case it is possi- ble to solve for enough points and centers using a digital computer so that the locii of all points can be plotted. These locus curves can then be used to select the mech- anism after a few trials. A direct method of selecting the best mechanism has not yet been devised. It should be possible, however, to write a computer program that will determine error over some set range for a finite number of the possible mechanisms so that the one with minimum error can be se- lected. The path generating method of synthesis involves a selection of the relationship between the moving plane angular displacement and the displacement of the tracing point on the coupler. The best relationship must be ob- tained by trial since, due to the nature of the problem of path tracing with a point, none will be given in the initial statement. It would seem that this is an area for further work. Obviously, a different rolling curve pair will result from each different angular displacement-path 78 displacement relationship. Once the selection is made, two other selections are necessary: first, the tracing point position so that equations of position may be writ- ten and second, the choice of the position angle 0. Once the tracing point position is fixed and the equations written, the angle (1 can be chosen either at random or a systematic exploration can be made using different values of a. As in the case of the function generator, a digital computer program can be written and used to obtain values for plotting the locus curves. These curves are very helpful in selecting the mechanism. The duplication of rolling curve plane motion as a means of substituting four bar linkages for rolling curves has been considered only briefly. The case included here, that of a disc rolling on a disc, was the most successful of all syntheses attempted in length of match. As pointed out earlier, it has an application where there is an in- centive to replace a pair of gears with a four bar link- age. The example shows the application where one gear is fixed. It is also possible to use the linkage to replace a pair of fixed center gears. Such use is a simplifica- tion of the function generator problem with constant an- gular velocity ratio. Other rolling curves in machines may be approximated with this technique. APPENDIX 80 COMPUTER PROGRAMS AND RESULTS The locus curves for the two path generating mech- anisms and the function generator were obtained using an IBM 1620 digital computer having a FORTRAN input. Since this is a widely used computer, the programs are included here. However, they are specialized programs and must be altered for other mechanisms. They are arranged to give the X and Y components of the points of stationary curv- ature for incremental values of a of 10°. Finer incre- ments can be obtained by rewriting the increment state- ment, A = A + 0.1745 (A is a ) to read the desired value. It is not necessary to index a around for 360°, because ml for a = 90° becomes m2 for a = 90°-I» 180°. Thus, if Aa(orAA) is taken to be one half of 0.1745 then the 100ping statement, DO 221 = 1,36 will supply enough points. Synthesis of other mechanisms can be obtained by rewriting the program statements as necessary to suit the new requirements. The form is suitable for specific mech- anisms. Specifically, the generation of a different func- tion would require that certain statements of table ‘7 be changed. These are: Nos. 11 and 12 for a different mat- ching point; 24 through 36 for a different function. 81 Statements 26 through 36 can be made general by replacing numerical values with alphabetical terms which are then defined earlier in the program. Table 5 FORTRAN PROGRAM FOR STRAIGHT LINE MECHANISMS i--’ OOGEQOWPWN i-’ FHA AHA STATIONARY CURVATURE POINTS AND CENTERS=STRAIGHT L. LINE MECHANISM A=0 DO 221=l,36 A=A+0.17h5 TC=COSF(A) TS=SINF(A) T82=SINF(2.0*A) QP1=0.25*(TC-TS+SQRTF(17.0-TS2)) QP2=0.25*(TC-TS-SQRTF(17.0-TS2)) YP1=QPl*TC YP2=QP2*TC XP1=QP1*TS XP2=QP2*TS Y1Pl=(l.0+XPl)/(l.O-YP1) Y1P2=(1.0+XP2)/(l.0-YP2) Y11Pl=§XPl-YPl-QP1**2)/(1.0-YP1**3) Y11P2= XP2-YP2-QP2**2)/(1.0-YP2**3) CX1=XP1-(Y1Pl*(1.0+YlPl**2))/Y11Pl CX2=XP2-(Y1P2*(l.0+YlP2**2))/Y11P2 CY1=YP1+(l.0+YlPl**2)/Y11Pl CY2=YP2+(1.0+Y1P2**2)/Y11P2 PUNCH 23,A,YP1,YP2,XP1,XP2,CX1,CX2,CY1,CY2 FORMAT(F7.4,8F9.3) END Symbols A =a_, xp1 = X91, YlPl = til, " YllPl = Ycl, 0x1 = 0x1, 0Y2 = 0Y2. 82 Table 6 FORTRAN PROGRAM FOR THE PARABOLIC PATH MECHANISM \OCDQOU'IPWNl-J PATH GENERATING MECHANISM-PARABOLA A=0.0 DO 29I=1336 A‘A+O.l7h5 TC=COSF(A-l.0) TS=SINF(A-l.0) (TC2=COSF(2.0*(A-1.0)) T32=SINF(2.0*(A-l.0)) D=(5.0+T32-3.0*T02)/(h.0*TC+5.0*TS) E=(7.0*TC+5.0*TS)/(4.0*TC-5.0*TS) GMl=0.l765*(-D+SQRTF(D**2.0+16.0*E)) GM2=0.1765*(-D-SQRTF(D**2.0+16.0*E)) XPl=l.0+GMl*l.hlh*TS XP2=1.0+GM2*1.A14*T3 YP1=2.0+GM1*1.111*T0 YP2=2.0+0M2*1.111*T0 YlPl=(1.0+GMl*1.hlA*TS)/(l.0-GM1*1.A1A*TC) Y1P2=(1.0+GM2*1.414*TS)/(l.0-GM2*1.41A*TC) Y11Pl=(-O.50+GM1*1.A14*(TS-O.50*TC+1.414*GM1))/ (1.0-GM1*1.114*T0)**3.0 Y11?2=(-0.50+0M2*1.411*(Ts-0.50*TC+1.411*GM2))/ (l.0-GM2*1.414*TC)**3.0 CXl=XPl~YlPl*(1.0+Y1Pl**2.0)/(Y11Pl) CX2=XP2~Y1P2*(1.0+Y1P2**2.0)/(Y11P2) CYl=YP1 + (l.0+YlP1**2.0)/(Y11Pl) CY2=YP2+(1.0+Y1P2**2.0)/(Y11P2) PUNCH 25,A,XP1,YPl,CXl,CY1,CX2,CY2 FORMAT(3F7.3,2F10.3,2F7.3,2F10.3) END Added symbols: GMl = ml, GM2 = m2 83 Table 7 FORTRAN PROGRAM FOR A FUNCTION GENERATOR - v,= 331°2 pmpowrwmw FUNCTION GENERATING MECHANISM - CASE ONE A=0.0 8-1.0 FI=A.0 D0 31I=1,36 A=A+0.1715 TH=B-FI+A TC=COSF(B) TS=SINF(B) TCA=COSF(A-FI) TSA=SINF(A-FI) TCB=COSF(TH) TSB=SINF(TH) TCB2=COSF(2.0*TH) TSB2=SINF(2.0*TH) D=(3.6+25.75*TSB2+O.72*TCB2)/(37.75*TCB+0.72*TSB) E=(1l.95*TCB-0.72*TSB)/(37.75*TCB+0.72*TSB) GM1=O.25* -D+SQRTF(D**2.0+16.0*E)) GM2=O.25* -D-SQRTF(D**2.0+16.0*E)) DYl=-(TC+GM1*(TSA-0.03A*TCA))/(TS+GM1*(TCA+0.03h*TSA)) DY2=-(TC+GM2*(TSA-0.031*TCA))/(TS+GM2*(TCA+0.034*TSA)) DDY1=-(1.0+GM1*(A.79*TSB-0.l905*TCB+3.79*GM1))/ (TS+GM1*(TCA-0.03A*TSA))**3.0 DDY2=-(1.0+GM2*(1.79*TSB-0.1905*TCB+3.79*GM2)i/ (TS+GM2*(TCA-0.034*TSA))**3.0 XP1=TC+GM1*0.2175*TSA XP2=TC+GM2*O.2175*TSA YP1=TS+GM1*O.2175*TCA YP2=TS+GM2*O.2175*TCA CXl=XP1~DY1*(1.0+DY1**2.0)/DDX1 CX2=XP2-DY2*(1.0+DY2**2.0)/DDY2 CYl=YPl+ 1.0+DY1**2.0)/DDY1 CY2=YP2+ l.0+DY2**2.0)/DDY2 PUNCH 32,A,XP1,YP1,CX1,CY1,XP2,YP2,CX2,CY2 FORMAT(3F7.3,2F10.3,2F7.3,2F10.3) END Symbols: A = a, B = 9: FI "¢: 9Y1: DYZ = Yilt Y£2 DDYl, DDY2 = Yzl, YEZ. BIBLIOGRAPHY BIBLIOGRAPHY Beggs, Joseph Stiles. Mechanism. New York: McGraw-Hill Book Co., 1955. Beyer, Rudolf. The Kinematic Synthesis of Mechanisms. Translated by H. Kuenzel. New York: McGraw-Hill Book CO., Inc., 1963. Cowie, Alexander. Kinematics and Design_0f Mechanisms. Scranton, Penna.:RInternationaI’TextbOOR 00., I961. Freudenstein, Ferdinand. "An Analytical Approach to the Desi n of Mechanisms." Transactions of the A§M§ v. 7 , (April, 195A), p. #83. Freudenstein, Ferdinand. "Approximate Synthesis of Four Bar Linkages." Transactions of the ASME, v. 77, (August, 1955), p. 853. Freudenstein, Ferdinand. "Structural Error in Plane Kinematic Synthesis." Egansactions of the ASME Journal of En ineerin for Induétr , v. 81, ser. B, No. I, (February, I959) p. 15. Freudenstein, Ferdinand, and Sandor, George N. "Synthesis of Path-Generating Mechanisms by Means of a Pgogrgmmed Digital Computer." ASME Paper No. 5 ”A- 50 Golber, H. E. "Rollcurve Gears." Transactions of the ASME, V. 61, (1939), p. 223. Hall, Allen 8., Jr. "Inflection Circle and Polode Curva- ture." Transactions of_thejFifth Conference on Mecganisms, Cleveland: The Penton Publishing CO., 195 . Hall, Allen 8., Jr. Kinematics and Linkage Design. Englewood Cliffs, N. J.: Prentice Hall, Inc., 1961. Hinkle, Rolland T. Kingmatics of Machines. Englewood Cliffs, N. J.: Prentice Hall, Inc., 1960. 85 86 Hirschhorn, Jeremy. Kinematics and Dynamics of Plane Mechanisms. NEw York: McGraw-Hill BOOK Co., Inc., 1962. Garnier, Rene. Cours de Cinematique. Tome III, Paris: Gauthier-VilIErs,‘l951. Martin, G. H. "Calculating Velocities and Accelerations in Four Bar Linkages." Machine Design, Cleveland: The Penton Publishing Co., Shaffer, B. W. and Cochin, I. "Synthesis of the Four Bar Mechanism when the Position of Two Members Is Pre- scribed." Transactions of the ASME, v. 76, (October, Williamson, Benjamin. An Elementary TreatiseL on the Differ- ential Calculus. ’LOndon: ‘Longmans, Green and Co., Ltd.,‘1927. Wilson, Edwin Bidwell. Advanced Calculus. Boston: Ginn and Company, 1911. Wolford, James C. and Haack, Donald 0. "Applying the Inflection Circle Concept." Transactions of the Fifth Conference on Mechanisms , C eveIand: The Penton Publishing Co., 1958. Yates, Robert C. A Handbook on Curves and Their Pro erties. Ann Arbor, MicH.: J. W. Edwaras, I957. ENGR L13. "'lll'llliilllii(lilflilliiiiiliiili’